Regarding your last sentence, you might investigate the book by G.A.Mikhailov titled
"Optimization of Weighted Monte Carlo Methods".
The originial is in Russian, but I believe it's been translated to English.
I don't own the book so I can't say precisely what's in it, but it's my understanding that it
considers (among other topics) statistical models (in this case vector Monte Carlo algorithms) for
solving systems described by integral equations.